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Here is a hard one from AoPS. I have given it my all, but need some fast help. Here it is:

Suppose that I have 6 different books, 2 of which are math books. In how many ways can I stack my  books on a shelf if I do not want the math books to be next to each other?

I tried to make 6 cases, a math book in the first position, in the second, third, and so on, but when I got to the third case, I found two more. It would get difficult quickly. Me and my parents don’t know what to do.

Feb 4, 2019

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The total possible arrangements = 6! = 720

Let's count the arrangements where the math books ARE together

They can appear in any of 5 positions and for each of these...they can be arranged in two ways....and in all these arrangements....the other 4 books can be arranged in 4! = 24 ways

So.....the total  arrangements where they are together = 5 * 2 * 4! = 240 ways

So....the arrangements where they DON'T appear together =  720 - 240  =   480

Feb 4, 2019
#2
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Thanks CPhill!

Guest Feb 4, 2019